In mathematics, the Fibonorial , also called the Fibonacci factorial, where is a nonnegative integer, is defined as the product of the first positive Fibonacci numbers, i.e.
where is the th Fibonacci number, and gives the empty product (defined as the multiplicative identity, i.e. 1).
The Fibonorial is defined analogously to the factorial . The Fibonorial numbers are used in the definition of Fibonomial coefficients (or Fibonacci-binomial coefficients) similarly as the factorial numbers are used in the definition of binomial coefficients.
Asymptotic behaviour
The series of fibonorials is asymptotic to a function of the golden ratio : .
Here the fibonorial constant (also called the fibonacci factorial constant) is defined by , where and is the golden ratio.
An approximate truncated value of is 1.226742010720 (see for more digits).
Almost-Fibonorial numbers
Almost-Fibonorial numbers: .
Almost-Fibonorial primes: prime numbers among the almost-Fibonorial numbers.
Quasi-Fibonorial numbers
Quasi-Fibonorial numbers: .
Quasi-Fibonorial primes: prime numbers among the quasi-Fibonorial numbers.
Connection with the q-Factorial
The fibonorial can be expressed in terms of the q-factorial and the golden ratio :
Sequences
Product of first nonzero Fibonacci numbers .
and for such that and are primes, respectively.
References
fr:Analogues de la factorielle#Factorielle de Fibonacci